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LP and Optimization Problems with Davison Orchards, Farmer Dave, Malmstrom Manufacturing, and BC Hyd

## Davison Orchards and the Production of Apple Butter, Apple Sauce, and Apple Jelly

Davison Orchards runs a family apple farm in Vernon, BC just a short drive from Kamloops. The bulk of the orchard apples are sold in boxes or baskets or go into the production of pies, tortes, and cobblers, for the Farmhouse Cafe. Davison has been getting request for products with a longer shelf life that don't require special handling (e.g., a freezer for apple pies). Apple butter, apple sauce, and apple jelly are three products that can have a one-year shelf life using sterilized jars and canning equipment.

Davison is winding down operations for the year and with COVID restrictions has about 52 bushels of unsold apples leftover. A bushel holds 125 apples. Davison doesn't have to incur any fixed costs to make the products — it can use its industrial kitchen. A 40-litre batch of Apple Butter will take 3.5 hours of cooking time. A 40-litre batch of Apple Sauce will take 3.0 hours,and Apple Jelly only take 2.5 hours of cooking time. A batch of Apple Butter will take an hour of labour time, three-quarters-of-a-hour for a batch of sauce, and jelly takes an hour-and-a-half of labour. Davison has 500 hours of cooking time and 250 hours of labour available. Butter, Sauce, and Jelly require 40, 55, and 20 apples respectively per batch. Davison estimates the revenues from these new longer shelf-life products will be 207, 177, and 155 \$/batch respectively.
1) Formulate an LP for Davison
2) Model the LP in Excel and solve using Solver
3) What is the revenue maximizing mix of apple products?
4) Davison has found a local farmer (Farmer Dave) who is willing to pay 7¢ per apple to take any unused apples to feed to his livestock. Davison estimates it would cost \$100 to bring the apples to the Dave. Is it worthwhile to get rid of any extra apples this way? Why? Why not?

Farmer Dave — Farmer Dave makes a cattle feed mix from four ingredients—oats, corn,soybeans, and a protein supplement — he may or may not add apples to the mix depending on what happened in the previous question J Farmer Dave has 300 kg of oats, 400 kg of corn, 200 kg of soybeans, and 100 kg of protein supplement available for the mix. In order to supply a balanced diet for the animals, Dave has the following requirements of his final feed mixture:
• At least 30% of the mix must be soybeans.
• At least 20% of the mix must be the protein supplement.
• The ratio of corn to oats cannot exceed 2 to 1.
• The amount of oats cannot exceed the amount of soybeans.
• The mix must be at least 500 kg.

## Farmer Dave and the Cattle Feed Mix

A kilogram of oats costs \$0.50, a kilogram of corn, \$1.20, a kilogram of soybeans, \$0.60, and a kilogram of protein supplement, \$2.00. Farmer Dave wants to know the number of kilograms of each ingredient to put in the mix to minimize cost. Like before ... Formulate an LP model for Farmer Dave and solve the model by using Excel Solver.

Sensitivity — Malmstrom Manufacturing produces ornate, decorative wood frame doors and windows. Each item produced goes through three manufacturing processes: cutting, sanding, and finishing. Each door produced requires 1 hour in cutting, 30 minutes in sanding, and 30 minutes in finishing. Each window requires 30 minutes in cutting, 45 minutes in sanding, and 1 hour in finishing. In the coming week Malmstrom has 40 hours of cutting capacity available, 40 hours of sanding capacity, and 60 hours of finishing capacity. Assume all doors produced can be sold for a profit of \$500 and all windows can be sold for a profit of \$400.
1. Formulate an LP model for this problem
2. Use Solver to Solve the LP. Produce a Sensitivity Report from the Solver dialog box.
3. Use the Sensitivity Report to answer the following questions...
a. If the profit on doors increased to \$700 would the optimal solution change?
b. If the profit on windows decreased to \$200 would the optimal solution change?
c. Explain the shadow price for the finishing process.
d. If 20 additional hours of cutting capacity became available how much additional profit could the company earn?
e. Suppose another company wanted to use 15 hours of Malmstrom’s sanding capacity and was willing to pay \$400 per hour to acquire it? Should Malmstrom agree to this? How (if at all) would your answer change if the company instead wanted 25 hours of
sanding capacity?

Anyhow... BC Hydro is responding to numerous calls for downed powerlines and power outages in the region. The downed wires create dangerous conditions for people living nearby, so BC Hydro wants to get the power restored safely as quickly as possible. They also want to do so in the most cost effective manner. Five sub-contractors to BC Hydro Arc, Buzz, Charge, Delta,and Electro provide crews to seven different areas of the lower mainland: North & West Van,Downtown, Vancouver South, Burnaby, New West & Surrey, Coquitlam, and Fraser Valley. The cost to send a crew to an area (in thousands of dollars) is calculated from union wages for workers, overtime premiums, mileage for call-outs, incidental expenses (e.g., putting a crew up in a hotel if the work will take longer than 12 hours), etc. In the table below for example, it will cost \$12,000 per crew to send a crew from Buzz to Burnaby. Buzz has 10 crews available. BC Hydro needs 8 crews working in the Burnaby area to clean up all of the downed lines.
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